On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature

TL;DR

This paper investigates the geometric structure at infinity of noncompact manifolds with nonnegative Ricci curvature and linear volume growth, proving they split off a line at infinity and exploring related geometric properties.

Contribution

It establishes that such manifolds always split off a line at infinity and provides new characterizations of linear volume growth, completing key steps in understanding isoperimetric sets.

Findings

Manifolds split off a line at infinity.

Level sets of Busemann functions have uniformly bounded diameter.

Limit spaces at infinity are cylinders with potentially nonhomeomorphic cross sections.

Abstract

We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes in the above setting. We also find that under our assumptions, the diameter of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when ends are collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic.